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September 28, 2009

This Week’s Finds in Mathematical Physics (Week 280)

Posted by John Baez

In week280 of This Week’s Finds, hear about the courses taught by Ashtekar and Rovelli at
the quantum gravity summer school in Corfu. Ashtekar spoke about
loop quantum cosmology, and how it could turn the Big Bang into a Big
Bounce. Rovelli spoke about spin foam models, and how the new
EPRL model cures many problems of the old Barrett-Crane model. The graviton propagator seems to work!

From the New Scientist:

Click on the picture to read the article!

By the way, this picture makes it look like there are galaxies near the Big Bounce in the ‘previous’ universe but not ours. I think a time-symmetrical picture is more likely. See the discussion below…

Posted at September 28, 2009 6:03 PM UTC

TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2068

Re: This Week’s Finds in Mathematical Physics (Week 280)

Thanks! I’m hoping that I can do a bit of work on quantum gravity now and then without getting too emotionally involved in it — it’s too much of a rollercoaster ride, and there’s too much fighting between different schools of thought. Math is great because you be quite sure you’re right. Physics is great because you can look up into the sky and see it. I want a balanced diet.

Re: This Week’s Finds in Mathematical Physics (Week 280)

which shows that something goes wrong when trying to “embed loop quantum cosmology into loop quantum gravity”. More precisely, it argues that:

The set of homogeneous isotropic connections, as used in loop quantum cosmology, forms a line ll in the space of all connections 𝒜\mathcal{A}. This embedding, however, does not continuously extend to an embedding of the configuration space l¯\overline l of homogeneous isotropic loop quantum cosmology into that of loop quantum gravity, 𝒜¯\overline{\mathcal{A}}. This follows from the fact that the parallel transports for general, non-straight paths in the base manifold do not depend almost periodically on ll. Analogous results are given for the anisotropic case.

Re: This Week’s Finds in Mathematical Physics (Week 280)

By the way, I would be very happy if anyone working on loop quantum cosmology could give me a intuitive physical explanation for the force that prevents the singularity.

My impression is the following, I’d be grateful to hear what is wrong about it.

What happens is a special case of this: start with a differential equation whose solutions diverge at the origin. Replace it with a difference equation on a subset of points that does not include the origin. This will have solutions not showing the singularity.

Is there anything deeper going on on LQC? I am seriously asking. I’d enjoy being educated here.

Because bounce solutions to gravity are of course well known, this has nothing to do with loop quantum gravity. It’s only that in LQG it is argued that one can remove the singularity by “stepping over it”.

Other people, under the headline “String cosmology” have argued that instead if there were a mechanism that would identify small scales with large scales – such as T-duality –, then that would give a rationale for arguing that the singularity in bounce solutions is not physical.

In both arguments I don’t see a particular “force” inducing the bounce. It’s more a kinematical effect in that one argues that kinematically the singularity is not part of the configuration space.

Re: This Week’s Finds in Mathematical Physics (Week 280)

Urs wrote:

My impression is the following, I’d be grateful to hear what is wrong about it.

What happens is a special case of this: start with a differential equation whose solutions diverge at the origin. Replace it with a difference equation on a subset of points that does not include the origin. This will have solutions not showing the singularity.

This may have been true for some early models — there’s been a lot of models between 2001 and now. It’s definitely not true for the currently popular models that Ashtekar was explaining.

It’s true that a differential equation is getting replaced by a difference equation. But the singularity is not being avoided simply by ‘stepping over it’. That would be a cheap trick In the models under study now, there really is an effective ‘force’ that prevents the singularity. And it arises from the mechanism I sketched:

Instead of talking about the curvature of spacetime at infinitesimally small scales, we can only measure curvature by carrying a particle around a finite-sized loop. This has little effect when spacetime is only slightly curved, but when the curvature is big it makes a big difference. This causes an effect very much like a force that prevents the Universe from crushing down to nothing as we follow it backwards in time.

A bit more precisely — but still quite roughly: if the curvature is xx, in loop quantum gravity we don’t have a quantum operator corresponding to xx. Instead we have something more like exp(ix)exp(i x). So we use something like sin(x)=(exp(ix)−exp(−ix))/2isin(x) = (exp(i x) - exp(-i x))/2i as a substitute for xx in certain equations. When the curvature is small these are almost the same. When it gets big, they’re different. This gives an effective force that prevents the singularity. This force starts getting big when the density is about 1/100 of the Planck density. When the density hits .41 times the Planck density, it’s enough to create a ‘bounce’.

(The math is actually a bit more complicated than what I said, but it’s similar in spirit.)

In fact, a bunch of calculations have shown that the quantum dynamics are quite nicely approximated by an ‘effective Friedmann equation’ — a differential equation like the usual classical one that describes the Big Bang, but with an extra term.

And, the solutions of the difference equation are very close to the solutions of this differential equation.

In Section 2, they start with the Friedmann equation for a homogeneous isotropic cosmology with k=0k = 0 and a massless scalar field ϕ\phi. In Section 2.1 they ‘deparametrize’ this equation, eliminating the time variable by using ϕ\phi as a clock field, or ‘internal time’ — a standard way of dealing with the problem of time in quantum cosmology. In Section 2.2 they review the standard Wheeler–DeWitt quantization of the resulting theory.

Then, in Section 2.3, they switch to loop quantization. They don’t give a vast amount of detail on why one should do this, but they do say what is being done. Briefly, instead of the variable c=da/dτc = da/d \tau (the rate of expansion of the universe, closely related to spacetime curvature in this model) we switch to using exp(ic)exp(i c) and exp(−ic)exp(- i c). In a deeper treatment this comes from using holonomies instead of the curvature at a point. This is what produces the effective force that prevents the singularity.

They don’t derive difference equations in this paper; instead, they just derive effective corrections to the usual Wheeler–DeWitt equation. Calculations have shown that solutions of the resulting differential equations closely match those of the difference equations that a more thorough approach yields. So, as far as physical intuition goes, perhaps the most important thing is to see where these effective corrections to the Wheeler–DeWitt equation come from, and why they prevent the singularity.

You’ll probably have more questions. I’m trying to recruit an expert on loop quantum cosmology to answer your questions. My own interests lie more in spin foams, rather than these phenomenological ‘loop-inspired’ models of cosmology. So, I’ll be much happier if someone else takes over here.

Re: This Week’s Finds in Mathematical Physics (Week 280)

Urs: for the level of detail you want, that paper won’t help. The other one I cited is better… but you might prefer Effective equations for isotropic quantum cosmology including matter by Bojowald, Hernandez and Skirzewski.

That paper makes no sense, whatsoever, to me.

To arrive at the mini-superspace approximation of LQC, we have integrated out an infinite number of modes. Why on earth should we expect the resulting effective Hamiltonian to take the absurdly simple form of their equation (22)?

Of course, we might expect that the true (disgusting, nonlocal in ϕ\phi, etc …) Hamiltonian might reduce to something simple in the regime of large FRW universes. But the LQC folks pretend to use it in the opposite limit, to say something about the big bang.

Re: This Week’s Finds in Mathematical Physics (Week 280)

Jacques wrote:

Why is that supposed to make sense?

I don’t know. I don’t think most people working on loop quantum cosmology have thought much about effective field theory. So, they aren’t thinking about ‘integrating out modes’. Instead, their work springs from the perhaps overoptimistic tradition of quantum cosmology, where you take a cosmological model with finitely many degrees of freedom and quantize it using the methods of quantum mechanics — not quantum field theory.

As you know (but others here will not, so I have to say it), when Wheeler and DeWitt first tried this, there were hopes that quantum effects would eliminate the Big Bang singularity. But they didn’t. So, people got excited when loop quantum cosmology, using somewhat different quantization methods, managed to eliminate the singularity. This led to a lot of discussion of differential equations versus difference equations, loop quantization versus Fock quantization, and so on — arguments about the validity of the methods whereby the loop quantum cosmology equations were ‘derived’. But personally I feel the word ‘derived’ is so out of place here that I’d be happy to ignore the ‘derivation’ and try to see what about the final equations winds up eliminating the singularity.
Or whether other assumptions could lead to effective Friedmann equations with this property.

But anyway: I’m not an expert on quantum cosmology, not interested in becoming one, not interested in debating the merits of different approaches, and completely unwilling to defend anyone’s work in this field.

Re: This Week’s Finds in Mathematical Physics (Week 280)

I was wondering whether there was any further work on the nature of the “area gap”. I mean, as symmetry-reduced versions of LQG, LQC models assume certain results directly from full LQG. For instance, the smallest nonzero area eigenvalue of LQG is the assumed step size (the so called “area gap”) of the LQC difference equation. For far this can be made more formally justifiable (on either physical and/or mathematical grounds)?

What is the connection of that problem (in terms of interpretation and consequences) with the fact that the Wheeler-de Witt equation can be made to agree with LQC (in the case of a k = 0, Λ = 0 FRW cosmological model, coupled to a massless scalar field), as the area gap diminishes (but taking in consideration that the approximation is not uniform in the chosen interval of “internal time”)? And also, with the fact that if the area gap is set to zero, LQC does not admit any limit, being an intrinsically discrete theory?

Re: This Week’s Finds in Mathematical Physics (Week 280)

Christine wrote:

For instance, the smallest nonzero area eigenvalue of LQG is the assumed step size (the so called “area gap”) of the LQC difference equation. For far this can be made more formally justifiable (on either physical and/or mathematical grounds)?

In the work that Ashtekar was summarizing, the step size is not equal to the area gap — but you’re right, it’s related.

I hope it’s clear why loop quantum cosmology is not my thing. Since we don’t really know the dynamics in loop quantum gravity, it takes a mixture of guts, caution, math, physical intuition, and trial and error to guess equations for loop quantum cosmology. This is a job for actual physicists. I’m more of a mathematical physicist. I like spin foams not only because they have a chance of providing loop quantum gravity with some dynamics that reduces to general relativity in a suitable limit, but also because they’re mathematically elegant, with relations to representation theory and category theory. Math like this makes me happy. Quantum cosmology mainly makes me nervous. Of course it’s fun to talk about, since it tackles big issues that everyone can enjoy, like the origin of the universe! But it’s not the sort of thing I’d enjoy working on, so I’m not an expert on it, and don’t plan to become one.

Re: This Week’s Finds in Mathematical Physics (Week 280)

A very concise and honest way to put it! I think that I understand what you mean. My case is much more serious than yours. I’ve always been “nervous” with physics in general. I have always seen it as some kind of “intrepid” adventure. Mathematics can be extremely difficult, but it is a question of building up things more and more from basic elements. It’s like a marathon running. Physics, at least at the frontier, is not like that. It is in a sense much more audacious and can make one nervous.

Re: This Week’s Finds in Mathematical Physics (Week 280)

Christine wrote:

I’ve always been “nervous” with physics in general. I have always seen it as some kind of “intrepid” adventure.

Yes, it is — and doubly so when the gap between theory and experiment is so enormous as in quantum gravity! Back in the golden age of particle physics you’d have all these mesons and hyperons with their own decay processes, and complicated nucleon-nucleon scattering amplitudes… it was an intrepid adventure to try to understand this stuff, but at least there were lots of numbers begging for explanation.

When I was younger the lure of quantum gravity was irresistible, but after a decade spent working on it with no clear sign of success I quit because I wanted to feel sure I’d discovered a truth or two. In math you can point to a theorem you’ve proved and say “that’s true”. But people like Ashtekar and Rovelli are incredibly courageous, because they’re willing to spend their whole career on theories that may never be confirmed in their lifetimes.

Re: This Week’s Finds in Mathematical Physics (Week 280)

Your observation is quite true.

But I wouldn’t mind to spend my whole career on quantum gravity if I had a permanent position (well, I do have, but not in that area), even with the perspective of missing a possible confirmation in my lifetime. One problem with this is, of course, the risk of getting “too far” with no experimental verification. I have no problem if one has a theory that offers predictions, even if it is beyond current technology. My problem is when people work with theories that can never be tested even in principle. I do not consider this scientific. But some people are willing to change the meaning of what is scientific in order to be able to say that they work on scientific theories. I think one should be extremely careful to designate what they are doing – philosophy, speculation, science, mathematics, and so on. But some people seem to become more and more used to living with the lack of experiment guidance, and the meaning of the scientific method is becoming strange to some.

Re: This Week’s Finds in Mathematical Physics (Week 280)

Now that things in spin foams are working nicely, where can you put particles in there?

I wouldn’t say that spin foams are “working nicely”: I just said that a few obstacles have been overcome. As I said:

Of course there are even bigger tests still ahead for this spin foam
model. We need to see if it reduces to general relativity in the
classical limit. In other words, we need to get Einstein’s equations
out of it. And we need to see if it reduces to the usual perturbative
theory of quantum gravity in some other limit. In other words, we
need to compute, not just graviton propagators (which describe the
probability of a lone graviton zipping from here to there on the
background of Minkowski spacetime), but graviton scattering amplitudes
(which describe the probability of various outcomes when two or more
gravitons collide).

Both these tasks are both computationally and conceptually
difficult. In other words, it’s not just hard to do the calculations:
it’s hard to know what calculations to do! When I said “in some
limit” and “in some other limit”, I know what limits
these are in a physical sense, but not how to describe them using spin
foams. Actually we seem closer to understanding graviton scattering
amplitudes, thanks to the work of Rovelli. But it seems miraculous
and strange that we can compute graviton propagators (much less scattering amplitudes) using very simple spin foams, as Rovelli and
his collaborators have done. Every time I meet him, I ask Rovelli
what’s going on here: how we can describe the behavior of a graviton
in terms of just a few 4-simplices of spacetime.

So, the road is still long, steep, and fraught with danger.

In particular, I would not be surprised if at some point we needed a spin foam model that includes matter to get everything to work out nicely. But we haven’t reached the point of being able to tell.

As for matter, I hope you know that in 3d quantum gravity, you get matter for free just by carving out ‘tubes’ in spacetime: they automatically act like worldlines of particles! This has been studied intensively by Barrett, Freidel, Baratin and collaborators — I gave a long list of references back in week222, and you really should read them.

In particular, these guys reached the point of being able to compute some Feynman diagrams for ordinary quantum field theory in 3d Minkowski spacetime using the G→0G \to 0 limit of the usual spin foam model for 3d quantum gravity!

Baratin and Freidel also know how to compute Feynman diagrams for ordinary quantum field in 4d Minkowski spacetime using a spin foam model. So this is a candidate for the G→0G \to 0 limit of a spin foam model containing gravity and matter.

Re: This Week’s Finds in Mathematical Physics (Week 280)

John wrote:

In particular, these guys reached the point of being able to compute some Feynman diagrams for ordinary quantum field theory in 3d Minkowski spacetime using the G→0 limit of the usual spin foam model for 3d quantum gravity!

I am not sure if one can use 3D as a toy model for more fundamental models. It seems gravitons don’t propagate in 3D because gravity is ricci flat. Plus weirdness happens in this kind of space time, like anyons obeying poincaré invariance, unlike in other dimensions where just integer or half integer are allowed.

Assume spacetime is an orientable smooth 4-manifold M.

If you are treating an extremely microscopic system, where quantum fluctuations are extreme, don’t you think you should find or not, exotic spheres first? Unlike the 3D case, where topological manifolds, smooth and piecewise connected are all the same and, mostly importantly, are unique, in 4D case this is not true. Top Man. is unique, but it is not the same as piecewise connected and smooth where the case is not settled. For example, if a exotic sphere is found, it won’t be unique, but more likely, will have an infinite number of non diff. types. Thus, one would lose an infinite quantity of states and a theory would not be reliable at microscopic level.

Re: This Week’s Finds in Mathematical Physics (Week 280)

I am not sure if one can use 3D as a toy model for more fundamental models. It seems gravitons don’t propagate in 3D because gravity is Ricci flat.

Right. 3d quantum gravity is an extremely dangerous toy model, because gravity is so different in 3 dimensions from higher dimensions.

I think we know very little about quantum gravity, and I have no desire to claim that any work that anybody is doing today is relevant to the real world — except a few very robust and simple calculations like Hawking radiation or the quantum gravity corrections to graviton propagators.

The great thing about 3d quantum gravity is that you can actually calculate things, and the theory keeps revealing new depths.

Treat it as pure mathematics, if you like, and you can learn to enjoy it.

Plus weirdness happens in this kind of space time, like anyons obeying Poincaré invariance, unlike in other dimensions where just integer or half integer are allowed.

Right. But in 4 spacetime dimensions you can have strings with exotic statistics. My paper with Alissa Crans and Derek Wise studies these in a 4d theory that’s analogous to 3d quantum gravity.

Again, think of it as pure math if you like. I’m making no claim that this is relevant to reality. I’m too old for that.

Daniel wrote:

John wrote:

Assume spacetime is an orientable smooth 4-manifold M.

If you are treating an extremely microscopic system, where quantum fluctuations are extreme, don’t you think you should find or not, exotic spheres first?

The sentence you quote came from an explanation of the Holst Lagrangian for general relativity! General relativity is a theory formulated on a smooth manifold. That’s why I wrote that sentence.

Re: This Week’s Finds in Mathematical Physics (Week 280)

Thanks for the paper on exotic statistics. But I have 2 questions:

1) “General relativity is a theory formulated on a smooth manifold. That’s why I wrote that sentence.” Yes, but these exotic manifolds are smooth. This is crazy. So, any deep discussion on general relativity should include them. I am confused why they are almost ever ignored, mainly in that because they are infinitely common.

Re: This Week’s Finds in Mathematical Physics (Week 280)

If someone knew something interesting to say about general relativity on exotic smooth manifolds, they would say it.

This is a different context, but have people thought about how this might affect topology change quantum gravity? (That is, it's not only a matter of topology change but also more generally diffeology change.)

Re: This Week’s Finds in Mathematical Physics (Week 280)

Thanks Daniel to take up the cudgels for exotic smoothness. Because of the equality between the combinatorical (or PL) and the smoothness structure, the results about exotic smoothness are very relevant for spin foam models as well. One uses triangulations of a smooth manifold and two inequivalent (in the sense of Pachner moves) triangulations are two inequivalent smoothness structures. You mentioned the topology change and its restriction by using smoothness. That is really relevant. For instance: it is forbidden (smoothly) to have a transition from a usual 3-sphere to the Poincare sphere.
I work for years on this topic (and wrote a book with Carl Brans). Currently we try to understand a relation between conformal field theory and exotic R^4 as well other structures like gerbes and orbifolds (see the papers gerbe paper and orbifold paper).

Re: This Week’s Finds in Mathematical Physics (Week 280)

I’d like to jump at the opportunity to do a little advertising for the book “Smooth Structures and Physics”: It’s style is clear and it does not assume too much previous knowledge, in fact it will be easy to read if you already know a bit about differential geometry and gauge theories (no, I’m not affiliated with the authors or publishers :-)

A qote and a question that could be if interest (p.269): “Finally, any quantum
gravity theory must have a classical limit which would include exotic four
manifolds. Thus, quantum gravity should incorporate exotic structures.”

Do you mean that in the classical limit there should be spacetime regions that look as if they had the canonical smooth structure and others that look like having an exotic one? If so, how would a transition from one to another look like to a classical observer?

Re: This Week’s Finds in Mathematical Physics (Week 280)

Yes, classically one can seperate exotic smoothness (in R^4) from each other. Thus, it is possible that there are exotic smoothness regions.
The effect of these regions was conjectured by Brans (some people call it the Brans conjecture): these regions are surrounded by an extra gravitational field. Later the conjecture was confirmed by Sladkowski and me. Such structures produce a gravitational lensing for instance.
Thats why we think that exotic smoothness is (probably) able to explain other exotic matter like dark matter or dark energy. Currently we are working on this.

Secondly, in my opinion exotic smoothness is so strongly connected with quantum gravity that a spacetime with exotic regions is not really classical.

“In 1854 B. Riemann, the father of contemporary differential geometry, suggested that the geometry of space may be more than just a mathematical tool defining a stage for physical phenomena, and may in fact have profound physical meaning in its own right [1]. But is it reasonable to contemplate to what extent the choice of mathematical model for spacetime has important physical significance? With the advent of general relativity physicists began to think of the spacetime as a differential manifold. Since then various assumptions about the spacetime topology and geometry have been put forward. But why should the choice of differential structure of the spacetime
manifold matter? Most of topological spaces used for modelling spacetime have natural differential structures and the question of their non-uniqueness seemed to be extravagant. Therefore, the counterintuitive discovery of exotic four dimensional Euclidean spaces following from the work of Freedman [2] and Donaldson [3] raised various discussions about the possible physical consequences of this discovery [4]-[21]. It has been shown that exotic (nonunique) smooth structures are especially abundant in dimension four [15] and there is at least a two parameter family of exotic R4s. Such manifolds play important role in theoretical physics and astrophysics and it became necessary to investigate the physical meaning of exotic smoothness.
Unfortunately, this is not an easy task…”

Re: This Week’s Finds in Mathematical Physics (Week 280)

Hi John,

I was thinking in how to make Reidemeister moves work in 4D. The thing is to avoid that a string (in the sense of the paper in question) trespass each other going to the 4th dimensions, I would had to glue together the 3D neighborhood of the contact points at each string.

If I imagine that:
* these strings are paths
* the contact point must be preserved
* that by inspecting the diagrams of the Reidemeister moves, one can define at that point of contact an order ing, and “above” and “below”

One must define locally a causal structure or a light cone, so that in the neighborhood of the point of contact one have ordering, the time, and that just 3 spacial dimensions are locally relevant.

So, could I say that the existence of such strings in 4D makes the existence of an everywhere local causal structure a necessary condition?

Re: This Week’s Finds in Mathematical Physics (Week 280)

Re: This Week’s Finds in Mathematical Physics (Week 280)

I’m glad someone is paying attention. The ‘downbeat’ Quote of the Week that now graces the end of week280 had formerly been my quote for week279. Since it’s about quantum gravity, I decided it would be more appropriate for week280. It should be read with a tinge of humor.

Originally this week’s Quote of the Week was:

I was sitting in a chair in the patent office in Bern when all of a
sudden a thought occurred to me. If a person falls freely, he will
not feel his own weight. - Albert Einstein

But I figure this should go on a Week devoted to classical general relativity!

My stock of good math and physics quotes is running low. Anyone know some more?

Re: This Week’s Finds in Mathematical Physics (Week 280)

Speaking of (experimental) theoretical physics,

apparently a team of European astronomers, including dr. Hongsheng Zhao and Dr. Benoit Famaey, have found a strange phenomenon regarding interactions between dark and normal matter.
Their article is going to be published in Nature today.

taken from newscientist:

”..Now, the tale has taken a deeper turn into the unknown, thanks to an analysis of the normal matter at the centres of 28 galaxies of all shapes and sizes. The study shows that there is always five times more dark matter than normal matter where the dark matter density has dropped to one-quarter of its central value.

The finding goes against expectations because the ratio of dark to normal matter should depend on the galaxy’s history – for example, whether it has merged with another galaxy or remained isolated during its entire existence. Mergers should skew the ratio of dark to normal matter on an individual basis.

“There is absolutely no rule in physics that explains these results,” says study co-author Hong Sheng Zhao of the University of St Andrews in the UK.

The authors suggest there may be an undiscovered force of nature working between the dark matter and the normal matter, since gravity alone cannot maintain this constant ratio…”

Re: This Week’s Finds in Mathematical Physics (Week 280)

Of course I was talking about ‘prevention of singularities’ in theories of loop quantum cosmology that are obtained by taking some ideas from loop quantum gravity — itself a speculative theory — and combining them with severe simplifying assumptions — namely, homogeneity and isotropy — in a rather controversial way. So, take everything I say here with a large grain of salt.

But, insofar as we can approximate the formation of a black hole by a process with perfect spherical symmetry, there’s hope that similar calculations can be done here too, and ‘singularity prevention’ will again occur. And indeed, there are calculations supporting this hope.

Also, people are starting to lessen the symmetry assumptions needed to get these results. For example, they’re working with Bianchi I cosmologies, which are anisotropic, and Gowdy spacetimes, which are even more complicated and less symmetrical.

So, Ashtekar is already dreaming of general singularity avoidance results.

Re: This Week’s Finds in Mathematical Physics (Week 280)

After the next ‘bounce’, should we expect from Ashtekar’s model(s) the same kind of fine-tuned universe as the one we are living in now (even if it isn’t the exact universe as ours)?

There probably won’t be a ‘next’ bounce.

It’s long been known that depending on the mass density of the universe and the cosmological constant, the universe may either keep expanding indefinitely or recollapse in a ‘Big Crunch’. Right now all the astronomical evidence points towards a universe that will expand indefinitely. If this is true, there’s just one Big Bounce.

One can adjust the mass density and cosmological constant in loop quantum cosmology and get models where there’s just one bounce. That’s the sort of model I was discussing — because that seems physically realistic. But one can also adjust these parameters to get models where the universe recollapses, and there are many big bounces, as shown in this computer calculation:

In this sort of scenario, it turns out the universe can bounce over 105010^{50} times before its wavefunction smears out significantly!

An interesting question is how the arrow of time works in this theory. Current research doesn’t answer that, because the model is too simple. But here’s an example of what I mean: in the one-bounce model shown below, does the bounce seem to lie in the past in both branches of the universe… or does it seem to lie in the past of one and the future of the other?

In other words, does this picture show two universes with a Big Bang in their past — or one with a Big Bang in its past and one with a Big Crunch in its future?

Personally I lean towards the former scenario, since a low-entropy state at the bounce is most likely to gain entropy in both directions. But this is the sort of thing where it would be good to do calculations.

Of course, if I’m right, the caption’s phrase “collapse of a preexisting universe” is a misleading way of describing what happened!

Anyway, we can rephrase your question as follows: are both branches of the above universe roughly the same? And if I’m right, the answer is yes.

Re: This Week’s Finds in Mathematical Physics (Week 280)

Vishal wrote:

The “effective force that prevents the singularity”, is that a property of matter or space-time or both?

Slightly tricky question.

It’s mainly the new treatment of spacetime that makes this force arise. The discreteness of spacetime at small distances is what makes loop quantum cosmology different from the standard approach to quantum cosmology. In the standard approach, the singularity is not avoided. In loop quantum cosmology, the discreteness modifies the equations in a way that becomes significant when the curvature of spacetime is big. This gives the effective force.

But, it’s crucial that the model includes matter as well as spacetime! It uses a very simple form of matter: a massless scalar field ϕ\phi whose value serves as a ‘clock’. In other words, we use this field to tell what time it is. We need some trick like this because of the famous ‘problem of time’ in quantum gravity.

Very crudely, the ‘problem of time’ is: how do you keep track of time in a world where everything is treated quantum-mechanically, even the geometry of spacetime itself?

Your next question might be: “What is this field ϕ\phi?” The answer is that we’re just playing around with very simple models here, which are not meant to describe our universe in much detail.
But as it happens, the most successful theories of inflation invoke a massless scalar field ϕ\phi which has been dubbed the ‘inflaton field’.

Re: This Week’s Finds in Mathematical Physics (Week 280)

Thank you for your replies. They were certainly very illuminating. If I could push on a bit (without risking getting lost or getting out of my depth), what do the models have to say about the “previous universe” (the one that presumably existed before the ‘bounce’)? In other words, how different was its geometry from the present one? (Of course, if I understand correctly, there’s just one universe, and that the use of words like “previous” and “current” for our universe need to be qualified. But, I hope the context here makes my question unambiguous.)

Re: This Week’s Finds in Mathematical Physics (Week 280)

I know I'm late, but I just now gave Week 280 the attention that it deserves. I noticed two things, quoting from near the beginning and very end of the web version:

about 1/100 of the Planck density

The rationalised (or reduced) Planck density is about 1/1001/100 of the (unrationalised, unreduced) Planck density, so this is just what I would expect. (More precisely, it's (8π)−2(8\pi)^{-2} of the Planck density, or about 1/6321/632 of the Planck length. I guess that that's more like 1/10001/1000 than 1/1001/100.)